Triality over schemes
type: Item
instance of: scholarly article
title: Triality over schemes
author: Cameron Ruether
published in: European Journal of Mathematics
publication date: 2024-12-13T00:00:00Z
review text: The article under review provides an alternative development of triality that does not use octonion algebras or symmetric composition algebras. Instead, the Clifford algebra of the split hyperbolic quadratic form of rank 8 and computations with Chevalley generators of groups of type \( D_{4} \) are used. The setting and notational system whithin which the article has been written can be found in [\textit{P. Gille} et al., ``Azumaya algebras and obstructions to quadratic pairs over a scheme'', Preprint, \url{arxiv:2209.07107}; \textit{C.~Ruether}, ``The canonical quadratic pair on Clifford algebras over schemes'', Preprint, \url{arXiv:2309.03077}]. In the paper, general sheaves are denoted by calligraphic letters, bold letters mean group sheaves, other rings and algebras are denoted by roman letters.\N\NThe preliminary part of the paper gives the setting over a scheme as well the definitions and background results for main objects studied in the paper such as quadratic triples, the groups \( \mathbf{Spin} ( \mathcal{A}, \sigma, f), \mathbf{O}^{+} ( \mathcal{A}, \sigma, f) \) and \( \mathbf{PGO}^{+} ( \mathcal{A}, \sigma, f) \), where \(( \mathcal{A}, \sigma, f) \) denotes an Azumaya algebra with quadratic pair over a field. The stack of trialitarian triples is defined and it is shown that it is equivalent to the gerbe of \( \mathbf{PGO}_{8}^{+} \)-torsors. It is shown also that it has endomorphisms generating a group isomorphic to the symmetric group \( \mathbb{S}_{3} \) and that several familiar cohomological properties of \( \mathbf{PGO}_{8}^{+} \) follow in this setting. Further, the author defines the stack of trialitarian algebras and shows that it is equivalent to the gerbe of \( \mathbf{PGO}_{8}^{+}\rtimes \mathbb{S}_{3} \)-torsors. For this reason, it is also equivalent to the gerbes of simply connected, respectively adjoint, groups of type \( D_{4} \). For a trialitarian algebra \( \mathcal{T}\), the author defines \( \mathbf{Spin}_{\mathcal{T}} \) and \( \m…
reviewed by: Evgenii L. Bashkirov
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 7958115
zbMATH Keywords: triality
zbMATH Keywords: \(D_4\)
zbMATH Keywords: quadratic pairs
zbMATH Keywords: Chevalley groups
zbMATH Keywords: octonion algebras
zbMATH Keywords: symmetric composition algebras
zbMATH Keywords: Clifford algebras
MaRDI profile type: Publication
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DOI: 10.1007/S40879-024-00771-Z
description: scientific article; zbMATH DE number 7958115